Question: $ A = \left[\begin{array}{rr}4 & 3 \\ 3 & 1\end{array}\right]$ $ F = \left[\begin{array}{rrr}-1 & 5 & -2 \\ 1 & -1 & 1\end{array}\right]$ What is $ A F$ ?
Solution: Because $ A$ has dimensions $(2\times2)$ and $ F$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A F = \left[\begin{array}{rr}{4} & {3} \\ {3} & {1}\end{array}\right] \left[\begin{array}{rrr}{-1} & \color{#DF0030}{5} & \color{#9D38BD}{-2} \\ {1} & \color{#DF0030}{-1} & \color{#9D38BD}{1}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{4}\cdot{-1}+{3}\cdot{1} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{4}\cdot{-1}+{3}\cdot{1} & ? & ? \\ {3}\cdot{-1}+{1}\cdot{1} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{4}\cdot{-1}+{3}\cdot{1} & {4}\cdot\color{#DF0030}{5}+{3}\cdot\color{#DF0030}{-1} & ? \\ {3}\cdot{-1}+{1}\cdot{1} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{4}\cdot{-1}+{3}\cdot{1} & {4}\cdot\color{#DF0030}{5}+{3}\cdot\color{#DF0030}{-1} & {4}\cdot\color{#9D38BD}{-2}+{3}\cdot\color{#9D38BD}{1} \\ {3}\cdot{-1}+{1}\cdot{1} & {3}\cdot\color{#DF0030}{5}+{1}\cdot\color{#DF0030}{-1} & {3}\cdot\color{#9D38BD}{-2}+{1}\cdot\color{#9D38BD}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-1 & 17 & -5 \\ -2 & 14 & -5\end{array}\right] $